Duplicated subsite

If the site is already cloned, you’ll have to select the posts from the source blog and use the “find unlinked children” to link to the posts on the clone (to prevent duplicates when broadcasting).

If you want to clone a site using NS Cloner, you can skip the above step by using the Broadcast NS Cloner add-on, that links during cloning.

Would the “Bulk Cloner” add-on replace NS Cloner in this scenario?

And while it’s a different question, does the “Cron Everywhere” add-on work if I set a server cronjob and turn off wp-cron in wp-config.php?

Yes, it would replace NS Cloner.

I’m not sure at all. The add-on works by scheduling a cron job, and in that cron job it then visits each site in the network and calling the cron URL.